A quadrilateral circumscribed about a circle has angles of 80,90,94, and 96. Find the measures of the four arcs determined by the points of tangency.

80,90,94, and 96 are in degrees.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Let the angles be `/_A=80^@,/_B=90^@,/_C=94^@,/_D=96^@` , and let the points of tangency be x(between A and B), y,z,w in order.

The measure of an angle formed by two tangents is equal to 1/2 the difference of the two arcs. Thus:





Let a=arc(xy),b=arc(yz),c=arc(wz) and d=arc(wx)

Then substituting we get:





We know that a=90. (A radius drawn to a point of tangency is perpendicular to the tangent; thus since angle B is 90 and the two radii to the points of tangent are 90 then the central angle is 90 and thus the arc is 90)

Then we have the following system:

160=90+b+c-d          70=b+c-d

180=-90+b+c+d        270=b+c+d

188=90-b+c+d          98=-b+c+d

192=90+b-c+d          102=b-c+d

Adding the first to the second yields 340=2b+2c or 170=b+c

Adding the third to the fourth yields 200=2d or d=100

Adding the second to the third yields 368=2c+2d or 2c=168; c=84

Then b=170-84=86.

Thus a=90,b=86,c=84, and d=100. These are the four arcs.

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial